By using the Dunford-Riesz functional calculus, to any sector
\( \Lambda \subset \mathbb { C}, \)  of the complex plane
we will associate a class class of elliptic operators \(\Psi ^m_{\rho ,\delta }(G\times \widehat{G};\Lambda )\)  as developed by the third author and J. Wirth in [1]}. There, one extended for any \(0\le \delta <\rho \le 1\)  the global functional calculus on compact manifolds due to Shubin [2]} under the restrictions \(0\le \delta <\rho \le 1\)  and \(0\le \delta <\rho \le 1.\)
